Advice on mathematics competitions

I greatly enjoyed my experiences with high school mathematics competitions (all the way back in the 1980s!). Like any other school sporting event, there is a certain level of excitement in participating with peers with similar interests and talents in a competitive activity. At the Olympiad levels, there is also the opportunity to travel nationally and internationally, which is an experience I strongly recommend for all high-school students.

Mathematics competitions also demonstrate that mathematics is not just about grades and exams. But mathematical competitions are very different activities from mathematical learning or mathematical research; don’t expect the problems you get in, say, graduate study, to have the same cut-and-dried, neat flavour that an Olympiad problem does. (While individual steps in the solution might be able to be finished off quickly by someone with Olympiad training, the majority of the solution is likely to require instead the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and so forth.)

Also, the “classical” type of mathematics you learn while doing Olympiad problems (e.g. Euclidean geometry, elementary number theory, etc.) can seem dramatically different from the “modern” mathematics you learn in undergraduate and graduate school, though if you dig a little deeper you will see that the classical is still hidden within the foundation of the modern. For instance, classical theorems in Euclidean geometry provide excellent examples to inform modern algebraic or differential geometry, while classical number theory similarly informs modern algebra and number theory, and so forth. So be prepared for a significant change in mathematical perspective when one studies the modern aspects of the subject. (One exception to this is perhaps the field of combinatorics, which still has large areas which closely resemble its classical roots, though this is changing also.)

In summary: enjoy these competitions, but don’t neglect the more “boring” aspects of your mathematical education, as those turn out to be ultimately more useful.

37 comments

Thanks for this kind of post. I am a college student from the Philippines (not a mathematics major though) and it is nice to hear from a highly-rated mathematician on things like these. Few top-rated mathematician will even try to post advices like these and they are invaluable to students like me in third world countries like us.

[…] also good to remember that professional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or […]

I know that mathematical competitions and mathematical research are very different, and that one who is good at the former may not be good at doing the latter. But what about the other way around. Do you think it is likely that (or have seen many examples where) one who is a good/creative researcher is not a particularly good problem solver (assuming that one has the knowledge to solve the problems and that time is not a constraint)?

Dear Prof Terrence Tao,
I’m a high school student in Vietnam. I decided to study Mathematics in higher level but I’m not sure if I’ve made a good choice or not. I’ve failed in my efforts to get into the Mathematics Gifted High School in my country, therefore I was unable to take part in any Mathematics competition. I found my passion in Mathematics very late but now I want to study Mathematics seriously. Do you think a “non-gifted” student like me should major in Mathematics?

[…] 3. “Boring” things are important … enjoy these competitions, but don’t neglect the more “boring” aspects of your mathematical education, as those turn out to be ultimately more useful. (source) […]

i think maths competitions are a great outlet for students to showcase their skills, and i am in awe of anyone who takes part (my maths is horrendous)! and not everyone is into sports or drama or other supposedly ‘cool’ activities so maths comps are great for those who enjoy maths and stats. thanks for a great post

Hey there, I was wondering if you could advise me on an issue about my capabilities in mathematics. I consider myself to be quite a good mathematician with a good mathematical intuition owing to me being rather good in normal maths lessons. However, I am terrible in maths Olympiads mainly because I get flustered and confuse myself (I never have any fun doing them unlike normal classes). This really damages my confidence as I fare far worse than my piers,many of whom spend much less time reading and thinking about maths. Do you think that my consistently terrible performances on maths challenges show that I am not the type of person for a career in mathematics or physics? I am sorry for such a silly self-absorbed question.

Very few mathematicians could recognize that classical math (Geometry, algebra, number theory) ‘inform’ modern math. Felix Klein spent his later part of his life promoting to ‘teaching Elementary Math from an Advanced Stand Point’.
As for Combinatorics, it appears in modern form in Hopf Algebra applied in Topology.

i don’t know what happen wit me..because when m goin study mathematics or science at that time i can able to undertand the difficulties or salution but when my teacher ask me that solutions the day of my study i cnt answer those questions..so please tell me sir how can i able to capture all those for all the time..

after you understand, practice speaking and writing out the solution and the difficult as if you were the teacher explaining to someone else. Then try to poke holes in your own explanations. That should help you retain what you already know, clarify what your weakness in understanding is, and help you be more confident that you really know what you think you know. Don’t be intimidated by your teacher – their job is to teach you, yours is to learn. If that isn’t happening, something isn’t right.

Dear prof.Terrence Tao, thank you for your intellectual advice about “mathematical competitions”. I live in Ethiopia & I have got MSc. Degree in Combinatorics but still i don’t have a published scientific paper. Please can you advice me about researches in Combinatorial Mathematics.

Hello Professor Tao, you probably already knew this, Australians Alex Gunning and Ishraq Huda have won IMO and IOI 2014 respectively! It’s great to see Australians excelling at more than just sport (to use your own turn of phrase). http://www.abc.net.au/news/2014-07-16/melbourne-teenage-mathlete-wins-gold-for-the-second-time-/5602226
I realise the Olympiads aren’t all there is to being a mathematician or a scientist; however, it’s pretty amazing to have two Aussies doing so well in the one year.
You remain a bit of an inspiration to us all down under!

competition is good at math because it make you to always practice so that you cannot lose..and when you are doing so it make you to be best student.I always do competition with my friend ,if i lose i becone angry that what make me to work hard so that i must not lose again.math is easy

We would be very grateful, if you send the information below to schools in your country.

We will be happy if they participate in our mathematical tournament and experience the pleasure to compete with children all over the world.

Dear friends of mathematics,

The Autumn round of the II International Tournament “Mathematics without Borders”, organized by Pedagogical Association EDUCATION WITHOUT BORDERS – Bulgaria and Science and Education Center INNOVATION – Kazakhstan, is over.
14 810 students from 20 countries (Azerbaijan, Afghanistan, Brazil, Bulgaria, France, India, Kazakhstan, Libya, Malaysia, Mexico, Moldova, Portugal, Romania, Russia, South Korea, Switzerland, Turkmenistan, Turkey, USA and Uzbekistan), located on three continents (Europe, Asia and America) took part in the contest.
To view the results, click here http://www.mathematicalmail.com/category.php?id=76

Medals for the winners and certificates for all participants were sent by mail.
Your photos from the award ceremony are welcome. If students and their parents do not object, we will post pictures on the website.
Those who could not participate in the first round, still have a chance to compete in the final round for the Tournament Cup – they can participate in the next two rounds – Winter (from 24 January to 1 February 2015) and Spring (from 21 to 29 March 2014). An invitation to participate in the final round will be sent to students with the highest sum of points obtained in two of the three remote contests.
For more information please visit our website: http://www.mathematicalmail.com
Thank you for your interest!
Thank you for your cooperation!
Congratulations to the winners!

[…] academic, physical, or intellectual development. Of course, one should still work hard, and participate in competitions if one wishes; but competitions and academic achievements should not be viewed as ends in […]

I have see so many oppoturnity but maths is like sport no need to be clever at school but you must love maths and practice it so tht u cn overcome theory but your thoughts ,and maths is also a key to better future and success

I find your articles quite inspiring and helpful. But I seem to be in a dilemma. I am a 15 year old boy from India, with an exceptional thirst for mathematics and physics. I have been very good in mathematics and my interest in the subject has led me to explore calculus. I have mastered differential calculus in 1 dimension and am currently reading up on integration. .. I have learnt quite a lot of topics in other fields of mathematics. … However my parents seem to think that mathematics does not have much scope for research…… This has led me to think on my choice for mathematics or physics. I would b grateful if you could provide your valuable advice

[…] Y in only Z years, or on scoring A on test B at age C..Of course, one should still work hard, and participate in competitions if one wishes; but competitions and academic achievements should not be viewed as ends in […]

Dear Prof Terry Tao,
Sir, are there any great mathematicians and professors who were not good at math Olympiads. I would be very much happy to know of some as I feel totally depressed by the so called ‘time limit’. Please help!!

[…] competition problems. (Note that competition math is fundamentally different to research math! See Tao, Evan Chen etc.) While I got silver with 36/100 points, the best two performances achieved 97/100 […]

It’s a lifetime memory when one achieves the success in any Olympiad Exams and gets praised throughout. Experience shared on the post is very well compiled and providing a brief information about Maths Olympiad Exam. Thanks for sharing the experience with us, keep updating us on the topics and latest developments in the field of Olympiad Exams.

Your sources motivates me even I’m 25th.I’m not very good at mathematics and computation. But I believe in beauty of mathematics, that is on the notion. Sometimes I browse your blog for some conflict intellectual questions in my Mind. Your books Analysis 1 and Analysis 2 motivates me to start from scratch and not worry about computational background.

For commenters

To enter in LaTeX in comments, use $latex <Your LaTeX code>$ (without the < and > signs, of course; in fact, these signs should be avoided as they can cause formatting errors). See the about page for details and for other commenting policy.